Former West Ham United striker Arnautovic opened the scoring in the 59th minute before former Brighton & Hove Albion player Mooy, who joined the club last month, came on to double their lead on his 30th birthday.
French midfielder Eddy Gnahore pulled one back for Wuhan 10 minutes from time to ensure a tight finish to the contest in the Suzhou hub, but Shanghai SIPG held their nerve to capture all three points.
Victory took the club to 26 points from 11 matches, with their only loss this season coming against Hebei CFFC.
Champions Guangzhou Evergrande were also winners on manager Fabio Cannavaro's birthday, as the Group A leaders staved off a late comeback attempt from Shandong Luneng to win 2-1 on Sunday thanks to goals from Paulinho and Elkeson.
Shandong's Wu Xinghan reduced the deficit in the 86th minute before a dramatic end to the contest saw both teams finish with 10 men left on the pitch.
Guangzhou Evergrande's Luo Guofu received his marching orders in the 90th minute for stamping an opponent while Shandong's Dai Lin was dismissed after a bad tackle.
Italian World Cup-winner Cannavaro's side now have 28 points from 11 games having won their last four on the trot.
Their City rivals Guangzhou R&F needed an 89th minute own goal and a 90th minute strike from Chang Feiya to snatch a point in a 3-3 draw with Jiangsu Suning in the Dalian hub.
Hebei also needed two late goals, including a penalty in the final minute to hold Beijing Guoan to a 3-3 draw while Shenzhen FC beat Henan Jianye 3-1.
Rafa Benitez' Dalian Pro were denied their third victory of the season when Shanghai Shenhua's Bi Jinhao struck with four minutes left to cancel out Salomon Rondon's first-half goal.
Qingdao Huanghai take on Chongqing Lifan on Wednesday while Tianjin Teda meet Shijiazhuang Ever Bright in a late game. (Reporting by Shrivathsa Sridhar in Bengaluru Editing by Christian Radnedge)